Regularization-based asset hedging tool

ABSTRACT

A regularization-based (RB) hedging tool identifies a recommended hedging portfolio that track a target asset and provides one or metrics indicating a predicted performance of the hedging portfolio relative to the target asset. The RB hedging tool uses a RB hedging model that is trained on price data from an observation period. Initial hyperparameters for the model are selected using asset price data from a validation period and the performance of the model is evaluated by applying it to asset price data from a backtest period. The end-user is presented with one or more metrics indicating the performance of the model and may modify one or more settings (e.g., hyperparameters) of the model. The model is retrained and reapplied to the backtest period, and the metrics are updated. Thus, end-users may tailor the model to their own particular needs and preferences.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/985,825, filed Mar. 5, 2020, entitled regularization-Based AssetHedging Tool, which is incorporated by reference.

BACKGROUND 1. Technical Field

The subject matter described relates generally to computer-basedsimulation tools and, in particular, to a regularization-based singleasset hedging tool.

2. Background Information

There are a wide range of scenarios in which it is beneficial to hedge atarget asset. Computer-based hedging tools automate or semi-automate theprocess of identifying hedging assets that track the performance of thetarget asset as closely as possible. Existing tools typically focus oncreating a hedging portfolio that matches past performance of the targetasset in a training period. This can result in overfitting, where thetool learns to reproduce noise or other unusual features in the trainingperiod, limiting the predictive power of such tools in generatingeffective risk hedging portfolios for the future performance of thetarget asset. Existing tools also focus on optimal weightings of eachasset in the hedging portfolio, requiring frequent rebalancing of thehedging portfolio. Such frequent rebalancing can result in significanttransaction costs. Another issue with many existing hedging tools isthat they are configured to solve specific, tightly constrainedproblems. Such tools use a model that is trained according to one ormore predetermined limitations (e.g., hyperparameters) and while theend-user can apply the model to different inputs, the limitations usedmay result in poor model performance for some inputs.

SUMMARY

A regularization-based (RB) hedging tool identifies a recommendedhedging portfolio that tracks a target asset and provides one or metricsindicating a predicted performance of the hedging portfolio relative tothe target asset. The RB hedging tool (the “RB hedger”) uses a RBhedging model that is trained on price data from an observation period.Initial hyperparameters for the model are selected using asset pricedata from a validation period and the performance of the model isevaluated by applying it to asset price data from a backtest period. Theend-user is presented with one or more metrics indicating theperformance of the model and may modify one or more settings (e.g.,hyperparameters) of the model. The model is retrained and reapplied tothe backtest period, and the metrics are updated. Thus, end-users maytailor the model to their own particular needs and preferences.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of a networked computing environment suitablefor providing a RB hedging tool, according to one embodiment.

FIG. 2 illustrates performance of a target asset and correspondinghedging portfolio in an observation period, a validation period, and abacktest period, according to one embodiment.

FIG. 3 is a block diagram of the RB hedger system shown in FIG. 1 ,according to one embodiment.

FIG. 4 is a flowchart of a method for using the RB hedging tool,according to one embodiment.

FIGS. 5A and 5B compare the performance of the RB hedging tool with aconventional hedging tool, according to one embodiment.

FIG. 6 illustrates the impact of the lasso hyperparameter on variousmetrics, according to one embodiment.

FIGS. 7A and 7B compare the holding error and transaction costs for theRB hedging tool and a conventional hedging tool, according to oneembodiment.

FIGS. 8A through 8C illustrate the impact of the lasso and ridge-likehyper parameters on various metrics, according to one embodiment.

FIG. 9 illustrate an example user interface for the RB hedging tool,according to one embodiment.

FIG. 10 is a block diagram illustrating an example of a computersuitable for use in the networked computing environment of FIG. 1 ,according to one embodiment.

DETAILED DESCRIPTION

The figures and the following description describe certain embodimentsby way of illustration only. It is noted that wherever practicablesimilar or like reference numbers are used in the figures to indicatesimilar or like functionality. Where similar elements are referenced bya common numeral followed by a letter, reference to the numeral alonemay refer to any such element or combination of elements, depending onthe context. One skilled in the art will readily recognize thatalternative embodiments of the structures and methods may be employedwithout departing from the principles described. For convenience, thedisclosed RB hedger is described in this disclosure in comparison to aconventional hedging tool referred to as “a conventional hedger.”

Overview

The RB hedger is based on a novel improved regularized optimization(e.g., a regression). In one embodiment, the regularized optimizationinvolves solving the following linearly constrained optimizationproblem:

$\begin{matrix}{{\min\limits_{s}{{{U \cdot s} - t}}_{2}} + {\lambda_{1}{s}_{1}} + {\lambda_{2}{{s \cdot p}}_{2}}} & (1)\end{matrix}$

where U is a matrix of time series of the returns of assets in theuniverse we are hedging with, t is the time series of returns of thetarget asset, s is the number of stocks in the hedging portfolio (thatis being solved for), and p is the vector of prices of assets in theuniverse at the end of the observation period.

More generally, a linear objective function corresponding to equation(1) can be written as:min_(t,p) ₁ _(,p) ₂ t+λ ₁ p ₁+λ₂ p ₂  (2)which can be optimized over linear and quadratic constraints.

An example linear constraint is defined for i=1, . . . , n (where n isthe length of the time series) as:

$\begin{matrix}{{{\sum\limits_{j = 1}^{p}( {A_{ij}x_{j}} )} - q_{i}} = b_{i}} & (3)\end{matrix}$where A is a matrix of historical prices of a given universe of assetsgoing back n days that includes p assets, x_(j) is the number of sharesin asset j at time i, q represents the optimization problem, and b_(i)is the target asset's notional at time i.

An example quadratic constraint is defined by the pair of conditions:

$\begin{matrix}{t^{2} \geq {\sum\limits_{i = 1}^{n}q_{i}^{2}}} & (4)\end{matrix}$ $\begin{matrix}{p_{2}^{2} \leq {\sum\limits_{j = 1}^{p}x_{j}^{2}}} & (5)\end{matrix}$

The RB hedger may apply different or additional constraints, such aslimiting the amount of any individual asset or class of assets (e.g.,assets within a business sector) included in the hedge portfolio. At ahigh level, the optimization problem for the RB hedger can be expressedby setting the objective function to be minimized (or maximized),defining the allowable ranges for each variable (e.g., whether it isstrictly positive or free, etc.), and setting the mathematicalconstraints (e.g., the linear and quadratic constraints each involvingseveral variables).

Referring again to equation (1), λ₁∥s∥₁ is called the lasso penalty andcontrols how concentrated the resulting hedge is (i.e. how many assetsthe hedging portfolio will include). The λ₂∥s·p∥₂ part of equation (1),a variation of the so-called ridge penalty, controls how diversified thehedge is in terms of the different sizes of the positions in the hedgingportfolio. The technical term “ridge” refers to using the squared

₂-norm ∥−∥₂ ² for regularization. In contrast, equation (1) uses theunsquared norm, and can thus be thought of as “Euclidean norm”regularization. Thus, it is referred to as ridge-like herein.

The amount of concentration and diversity is controlled by the values ofthe λ₁ and λ₂ hyperparameters. Optimal values for these parameters maybe estimated by empirical means. For example, estimates of the optimalparameters may be determined by measuring the holding error for variouscombinations of values and picking the combination with the best results(“grid search”).

Unlike previous methods, the lasso and ridge-like penalties may preventthe hedge from overfitting the past by causing the solution to settle ata compromise between a concentrated and a diversified hedge.Furthermore, in some embodiments, the end user may be able to vary theconcentration and diversity hyperparameters at run time and quickly viewthe resulting variations in the calculated hedging portfolio and itsperformance relative to the target asset with regard to one or moremetrics. Thus, end users may tune the model to their individual needsand preferences substantially in real time without requiring the modelprovider to update the model.

As a result of these enhancements, the novel RB hedger may provide thefollowing advantages in single asset hedging:

-   -   Concentration: This corresponds to focusing a hedge on a reduced        number of assets in the portfolio that is constructed. Higher        concentration means fewer assets are present in the hedge        portfolio.    -   Diversity: Increasing uniformity in the distribution of weights        in the hedge portfolio. Higher diversity means more balance in        the distribution of weights for assets in the portfolio.    -   Reduced transaction costs: Decreasing the cost of trading the        positions in the hedge.

Generally, a good single asset performance hedge strikes a balancebetween concentration (it includes only a few assets that track theperformance of the target asset) and diversity (it puts approximatelyeven weights on the selected assets). The RB hedger uses a novelformalization of this idea, in a way that is highly interpretable andcustomizable, and improves the accuracy of the resulting hedges.

In one embodiment, the RB hedger uses an interface that binds its nativeprogramming language to Java® to achieve significant latencyimprovements over existing hedging tools. This may enable the RB hedgerto rapidly refit and optimize the model in a matter of seconds (e.g., inless than one second, less than three second, less than five seconds,etc.) any time arguments are modified by the end user. To achieve suchrapid refitting and optimization of the model, the RB hedger makes callsto the interface, which wraps the optimizer's Java® api, in oneembodiment. In other embodiments, other language-specific APIs may beused. Using the interface in this way, the underlying task and solutionobjects may be parallelized “in process” when solving the objectivefunction for each constraint. Furthermore, the hyperparameter search maybe more effectively distributed. These improvements provided by theinterface may result in a speed increase of up to five times relative toconventional approaches when averaged across various inputs.

Example Systems

FIG. 1 illustrates one embodiment of a networked computing environment100 suitable for providing a RB hedging tool. In the embodiment shown inFIG. 1 , the networked computing environment includes a RB hedger system110 and a set of client devices 140, all connected via a network 170. Inother embodiments, the networked computing environment 100 includesdifferent or additional elements. In addition, the functions may bedistributed among the elements in a different manner than described. Forexample, the RB hedger may be provided by a stand-alone system thatperforms the functionality of both the RB hedger system 110 and a clientdevice 140.

The RB hedger system 110 includes hardware and software for implementingthe RB hedger. In various embodiments, the RB hedger fits a hedge over auser-defined observation period in a way that is more likely to make itmore accurate in the future. In particular, it does so by using both anobservation period to compute the hedge and a validation period toensure that the hedge has not overfit to the historical data.

Generally speaking, a simple linear regression will appear more accuratethan the RB hedger in the observation period. This is, however, anindication that overfitting may occur. That is, the simple linearregression does not generalize the lower holding error in theobservation period to the backtest period. In other words, the modellearns the noise of a dataset better than the signal that corresponds tobetter accuracy in new contexts. In the case of a conventional hedger,since no hyperparameters are used and a different optimization problemis being solved, the model is prone to overfitting theobservation/backtest data.

For the RB hedger, the opposite is true: the RB hedger is closer to thetarget asset in the backtest period rather than the observation period.Overfitting is reduced since the model is able to train, validate, andtest based on the use of hyperparameters that introduce bias into thetraining that reduce the model's tendency to focus on noise in theobservation period. This is the mechanism through which the RB Hedgermay improve accuracy: it learns to ignore noise during the past (theobservation period), thus making it perform better in predicting thefuture or a simulated future (the backtest period). Various embodimentsof the RB hedger system 110 are described in greater detail below, withreference to FIG. 3 .

FIG. 2 illustrates the performance of a target asset and correspondinghedging portfolio in an observation period 210, a validation period 220,and a backtest period 230, according to one embodiment. In the exampleshown, the observation period 210 is four months long, but other lengthsof observation period 210 may be used. This is followed by a shortervalidation period 220 (e.g., 1 month) and a backtest period 230 (e.g.,one month). In FIG. 2 , the observation, validation, and backtestperiods are shown as occurring in immediate succession. However, theremay be a gap between periods.

The observation period 210 is a specified date range of the price curvesfor the core portfolio and hedge to use in the optimization. Theobservation period 210 corresponds to the period in which the RB hedgeroptimization is “fit”, where the target number of shares for each assetin the hedge is found, and represents the “training period” (in machinelearning terminology). In a conventional hedger, this corresponds to theperiod in which the optimal weights are found for each constituent inthe constructed hedge portfolio. The observation period 210 may beselected by the user. Unlike a conventional hedger, in the case of theRB hedger, a small period of time at the end of the observation period210 is replaced with the validation period 220.

The validation period 220 is a date range in which the model used by theRB hedger is validated to find optimal (or at least approximatelyoptimal) hyperparameter values, which are used in the optimizationproblem to increase accuracy in the “simulated out-of-sample period.”This period may be automatically constructed by the RB hedger and maydepend on the size and dates of the observation period 210 specified bya user. For example, if a conventional hedger would use an observationperiod 210 of one year from Oct. 18, 2018 to Oct. 18, 2019, the RBhedger may choose its observation period 210 to be Oct. 18, 2018 to Sep.18, 2019 and its validation period 220 to be Sep. 19, 2019 to Oct. 18,2019. In other embodiments, different length of validation period 220may be used.

The backtest period 230 is a date range in which the backtest of thehedge is run. In the case of a conventional hedger, this may be the sameas the user-specified observation period 210. A conventional hedgermodel may fit its linear model over the observation/backtest period andtry to minimize the holding error in this period. In the case of the RBhedger, the backtest period 230 is also known as the “simulatedout-of-sample” or “test” period in which it is measured how closely theRB hedger is tracking the holding error of a target asset over a daterange after the observation/validation periods. This period may bechosen by the user after the observation/validation periods aredetermined. In one embodiment, the range for this period must be greaterthan 22 days. In other embodiments, the possible lengths of the backtestperiod 230 may be different.

Referring again to FIG. 1 , the client devices 140 are computing devicescapable of receiving user input as well as transmitting and receivingdata via the network 170. The client devices 140 can take various formssuch as desktop computers, laptop computers, personal digital assistants(PDAs), mobile telephones, smartphones, and other suitable devices. Inone embodiment, a client device 140 provides an interface (e.g., awebpage presented in a browser, an app, etc.) with which users mayinteract with the RB hedger system 110. The user provides parameters andrequests generation of a hedging portfolio for a specified asset. Theclient device 140 may also present information to the user about theresulting hedging portfolio generated by the RB hedger system 110.

The network 170 provides the communication channels via which the otherelements of the networked computing environment 100 communicate. Thenetwork 170 can include any combination of local area and/or wide areanetworks, using both wired and/or wireless communication systems. In oneembodiment, the network 170 uses standard communications technologiesand/or protocols. For example, the network 170 can include communicationlinks using technologies such as Ethernet, 802.11, worldwideinteroperability for microwave access (WiMAX), 3G, 4G, 5G, code divisionmultiple access (CDMA), digital subscriber line (DSL), etc. Examples ofnetworking protocols used for communicating via the network 170 includemultiprotocol label switching (MPLS), transmission controlprotocol/Internet protocol (TCP/IP), hypertext transport protocol(HTTP), simple mail transfer protocol (SMTP), and file transfer protocol(FTP). Data exchanged over the network 170 may be represented using anysuitable format, such as hypertext markup language (HTML) or extensiblemarkup language (XML). In some embodiments, all or some of thecommunication links of the network 170 may be encrypted using anysuitable technique or techniques.

FIG. 3 illustrates one embodiment of the RB hedger system 110. In theembodiment shown, the RB hedger system 110 includes a user interfacemodule 305, an observation module 310, a validation module 320, abacktest module 330, and a datastore 340. In other embodiments, thenetworked computing environment 100 includes different or additionalelements. In addition, the functions may be distributed among theelements in a different manner than described.

The user interface module 305 provides a user interface for display toan end user (e.g., at a client device 140). In one embodiment, the userinterface includes controls to enable the end user to select a targetasset for which a hedge portfolio is desired and the values to use forhyperparameters (e.g., concentration and diversity). Additionally oralternatively, the user interface may include controls to enable the enduser to select one or more of the observation, validation, or backtestperiods to use. The user interface may also include one or more metricsindicating the performance of the calculated hedge portfolio relative tothe target asset in the form of numerical values, charts, graphs, andthe like. In other embodiments, some of the parameters may be hardcodedor determined from other parameters selected by the user. For example,the user interface may enable the end user to select the observationperiod 210 and the one-month period after the selected observationperiod 210 may be automatically selected as the validation period 220.

If the end user changes a hyperparameter (or defines a differentobservation, validation, or backtest period), the RB hedger system 110may automatically determine an updated hedge portfolio and update one ormore metrics displayed in the user interface to indicate the performanceof the newly calculated hedge portfolio. Thus, end users can tweak thehyperparameters according to their needs or preferences and quickly seethe results on the hedge portfolio. The user interface may also includecontrols to enable the end user to obtain the calculated hedge portfolio(e.g., by initiating trades for assets in an electronic trading system).

The observation module 310 fits the RB hedger model to the price of thetarget asset over the observation period 210. The observation module 310returns a portfolio of assets that tracks the performance of the targetasset as closely as possible.

The validation module 320 chooses values for the model hyperparametersbased on the price of the target asset during the validation period 220.The hyperparameters are tunable model parameters that are tuned with thegoal of optimizing the resulting hedge. In one embodiment, thehyperparameters optimize the resulting hedge in the sense that theselected values minimize holding error for the hedge.

The hyperparameters are tunable coefficients in the RB hedger thatcontribute to total model concentration and diversity. In oneembodiment, the hyperparameters include lasso and ridge-likehyperparameters, as described previously. Since the RB hedger solves forbuy-and-hold, the validation module 320 makes use of the holding errorin the optimization problem as a measure of accuracy. The holding erroris the standard deviation of the difference between the hedge and targetreturns over the specified time interval (i.e., different forobservation, validation, and backtest periods) without hedgerebalancing. In contrast, a conventional weights-based hedger may assumeconstant rebalancing and this seeks to minimize the tracking error,which takes into account hedge rebalancing.

To minimize the holding error for the RB hedger, the validation module320 considers a grid of possible combinations of lasso/ridge-like valuesusing a grid search algorithm run in the validation period 220. The gridsearch chooses the optimal pair of hyperparameters (lasso/ridge-likevalues) that minimize the holding error during this period. The selectedhyperparameters are then used to refit the linear model in theobservation period 210 and used in the backtest period 230 to measureholding error. This approach optimizes the RB hedger for a holdingperiod equal to the length of the validation period 220 (e.g., one monthby default).

The backtest module 330 measures how closely the RB hedger is trackingthe performance of the target asset over a date range after theobservation/validation periods. This date range corresponds to the“simulated out-of-sample” period. The RB hedger is aiming to maximizeperformance in this period. In other words, the goal is for the hedgerto track the target asset as closely as possible during this periodsince it simulates the period during which investors will actually holdthe calculated hedge that the RB hedger outputs.

The datastore 340 includes one or more computer-readable media thatstore data and software used by the RB hedger system 110. For example,the parameterized model and data sets of the prices of the target assetand assets in the hedging portfolio may all be stored in the datastore.Although the datastore 340 is shown as a single entity within the RBhedger system 100, it may be divided into multiple parts, some or all ofwhich may be accessed remotely (e.g., via the network 170).

Example Method

FIG. 4 illustrates a method 400 for using the RB hedger to identify ahedging portfolio for a target asset, according to one embodiment. Thesteps of FIG. 4 are illustrated from the perspective of the RB hedgersystem 110 performing the method 400. However, some or all of the stepsmay be performed by other entities or components. In addition, someembodiments may perform the steps in parallel, perform the steps indifferent orders, or perform different steps.

In the embodiment shown in FIG. 4 , the method 400 begins with the RBhedger system 110 receiving 410 a selection of a target asset. Forexample, a user may select a desired target asset via a user interfacedisplayed at a client device 140 and the selection may be sent to the RBhedger system 110 via a network 170. The system may also receive userselection of an observation period 210, a validation period 220, and abacktest period 230. Alternatively, default time ranges for one or moreof these periods may be used.

The RB hedger system 110 trains 420 the RB hedging model based on pricedata for the target asset and potential hedging assets during theobservation period 210. The RB hedger system 110 may choose 430 initialhyperparameters for the model based on price data for the assets duringthe validation period 220. In one embodiment, the initialhyperparameters are chosen 430 to minimize the holding error of thehedging portfolio. The RB hedger system 110 applies 440 the model toprice data for the assets during the backtest period 230.

The RB hedger computes the optimal number of shares for each asset inthe hedging portfolio and presents 450 the results to the user. In oneembodiment, the results presented 450 to the user include the assetsthat makeup the hedging portfolio and one or more performance metrics,such as the holding error, tracking error, etc. One advantage of thisapproach it that the hedging portfolio is assumed to be held throughoutthe backtest period 230 (or for a substantial portion of it), withoutrebalancing it daily. This is in contrast to previous methods, whichcompute the optimal weights for each asset in the hedging portfolio,rather than the optimal number of shares. The disadvantage of theseprevious methods is that they provide a solution that can be realizedonly by regularly (e.g., daily) rebalancing the hedging portfolio daily,incurring additional transaction costs. As an example, say the hedgesolution is 50% asset A and 50% asset B in the weights formulation. Tomaintain the weighting of 50% for each asset, the portfolio must berebalanced daily (because asset A could increase in price and asset Bcould decrease in price, where you now have 51% asset A and 49% asset Bin your portfolio). Whereas if the solution is 50 shares of A and 50shares of B then you will still maintain 50 shares of each asset even ifthe prices increase or decrease.

Once the results have been presented 450 to the user, the user maymodify one or more settings of the model (e.g., the values of one ormore hyperparameters) via the user interface. When the RB hedging system110 receives an updated setting, it retrains the model and presents 450updated results to the user. For example, the model may be refitted tothe observation and validation period using the updated settings andreapplied to the backtest period to generate one or more performancemetrics for the retrained model. Thus, the user can explore howdifferent choices (e.g., different values of the diversity andconcentration hyperparameters) impact the makeup and performance of thecalculated hedging portfolio.

The user interface may also include controls to enable the user toinitiate 460 trades to acquire the hedging portfolio calculated by theRB hedging tool. For example, once the user has experimented withdifferent settings and a hedging portfolio that meets the user's needsand preferences is displayed in the user interface, the user may selecta button or other control to automatically place trades with anelectronic trading system to obtain the currently displayed hedgingportfolio.

Example Use Case

FIG. 5A illustrates the performance of a conventional hedging tool. FIG.5A includes a pair of performance curves (in terms of percent change inasset returns) for the conventional hedger applied to a target asset ina backtest period 230 (between May 21, 2018 and Jun. 29, 2018). Thehedge line is tracking the core (target asset) line, attempting tominimize the tracking error.

FIG. 5B illustrates the performance of one embodiment of the RB hedger.FIG. 5B includes a pair of performance curves (in terms of percentchange in asset returns) for the RB hedger applied to the target assetin the backtest period 230 (between May 21, 2018 and Jun. 29, 2018). Thehedge line is tracking the core (target asset) line, attempting tominimize the holding error. Note that the tracking and holding errors inthe backtest period 230 are both lower for the RB hedger compared withthe conventional hedger.

FIG. 6 illustrates the impact of the lasso hyperparameter on variousmetrics, according to one embodiment. The metrics include transactioncost, tracking error, holding error, and number of assets generated forthe target asset in the backtest period 230. This illustrates thetradeoffs that a user can choose between when toggling differenthyperparameter values on the user interface for the RB hedger. In thiscase, the ridge-like hyperparameter is fixed at ten to illustrate thechanges in metrics that occur as tee lasso hyperparameter varies fromzero to ten. However, a user can choose to modify both of thesehyperparameters to fit their needs.

FIGS. 7A and 7B illustrate the performance of one embodiment of the RBhedger relative to a conventional hedger. The RB hedger was backtestedin various ways to validate its performance. The results indicate thatthe RB hedger increases the accuracy of the conventional hedger,especially for hedges computed over larger universes of assets. Thebacktests were run on a representative set of hedges similar to realhedges. For each of these hedges, the accuracy of a simple linearregression was compared to the accuracy of the RB hedger the monthfollowing the calculation (i.e. during the backtest period 230).

The results are for a sample of 101 hedges (101 unique targets, randomchoice of notional, observation period 210, maximum return deviation,maximum daily volume, and using assets from the Russell 3000 index toconstruct the hedge). FIG. 7A compares the holding error of the hedgesgenerated by the RB hedger and the corresponding hedges generated by theconventional hedger. The RB hedger had lower holding error for most ofthese random hedges. FIG. 7B compares the transaction costs for thehedges generated by the RB hedger and the corresponding hedges generatedby the conventional hedger. The transaction cost used in this case isthe weighted average estimated cost of trading the positions in thehedge with Goldman Sachs. The RB hedger had lower transaction costs formost of these random hedges.

FIGS. 8A through 8C illustrate the impact of the lasso and ridge-likehyperparameters on various metrics for one embodiment of the RB hedger.The RB hedger was backtested for 150 random target assets to compare howvarious metrics vary the values of the lasso and ridge-likehyperparameters are changed. These metrics include: transaction cost,tracking error, holding error, number of assets in hedge portfolio, andstandard deviation of weights of the assets in the hedge portfolio. Theresults were analyzed by considering a range of hyperparameter values,then averaged out each metric and plotting the averages. These plotsgive an indication of the behavior of the fitted hedges as thehyperparameters are varied.

In particular, FIGS. 8A through 8C illustrate average metrics generatedfrom 782 hedges (150 unique targets, random choice of notional,observation period 210, maximum return deviation, maximum daily volume,and a random choice of universe between S&P500, Russell 3000 or Nasdaq100). The average value for each metric is computed based on all of thehedges grouped by the ridge-like hyperparameter used in that hedge.

FIG. 8A illustrates average transaction cost, tracking error, holdingerror, and number of assets for variations in the lasso hyperparameter.The ridge-like hyperparameter is fixed to be zero. FIG. 8B illustratesaverage transaction cost, tracking error, holding error, and number ofassets for variations in the ridge-like hyperparameter. The lassohyperparameter is fixed to be zero. FIG. 8C illustrates averagetransaction cost, tracking error, holding error, and standard deviationof weights for variations in the ridge-like hyperparameter. The lassohyperparameter is fixed to be zero. These plots indicate how thehyperparameters may be adjusted to achieve different objectives. Forexample, a low holding error may be obtained using high ridge-likevalues and low lasso values, but this will also increase the number ofassets in the hedging portfolio, which may be undesirable.

Example User Interface

FIG. 9 illustrates an example user interface 900 for the RB hedgingtool, according to one embodiment. In the embodiment shown, the userinterface 900 includes a target asset box 902, an amount box 904, anobservation period box 912, a validation period box 914, a backtestperiod box 916, a concentration slider 922, a diversity slider 924, ametrics display area 930, and an obtain hedge button 940. The depicteduser interface 900 is simplified to illustrate various aspects of the RBhedger tool. In other embodiments, the user interface 900 may includedifferent or additional elements.

The target asset box 902 identifies the currently selected target asset.The user may click on or otherwise select the target asset box 902 andchoose a different target asset. For example, the user may type the nameof an asset, selecting an asset from a drop-down list of availableassets, or provide a stock ticker for an asset, etc.

The amount box 904 indicates the amount of the target asset to be hedgedby the hedging portfolio. In FIG. 9 , the amount is indicated as adollar value. However, the amount may be indicated in other ways, suchas values in different currencies or a number of instances of the asset(e.g., a number of shares).

The observation period box 912, validation period box 914, and backtestperiod box 916 display the currently selected observation period,validation period, and backtest period, respectively. The user mayselect these boxes to modify the corresponding period. The user maydefine a period in any appropriate way, such as typing a beginning andend date or selecting a range of dates from a popup calendar. In someembodiments, one or more of the periods may be determined automaticallyand not editable by the user. For example, default periods such as thepreceding month for the backtest period, the month before that for thevalidation period, and the four months before that for the observationperiod may be used. As another example, the user may select one period(e.g., the observation period) and the other periods may beautomatically determined based on the user's selection (e.g., thevalidation period may always be a one-month period immediately after theobservation period).

The concentration slider 922 and the diversity slider 924 enable theuser to modify the values of the concentration and diversityhyperparameters, respectively. It should be appreciated that other typesof control may be used to modify the hyperparameters. It should also beappreciated that controls for modifying other hyperparameters may beincluded in the user interface 900, depending on the underlying modelbeing used by the RB hedger.

The metrics display area 930 includes information about the hedgingportfolio generated by the RB hedger using the values provided elsewherein the user interface 900. In one embodiment, the metrics display area930 displays values for one or more metrics indicating the performanceof the generated hedging portfolio relative to the target asset duringthe backtest period. The metrics may include the tracking error, theholding error, the daily correlation, the hedge transaction costs, theannual volatility, charts indicating the relative prices of the hedgingportfolio and the target asset, and the like. If the user updates anysettings of the RB hedger (e.g., by changing the value of one of thehyperparameters), the hedging portfolio is recalculated and the metricsdisplayed in the metrics display area 930 may be updated automatically.

When the metrics displayed in the metrics display area 930 meet theuser's needs and preferences, the user may select (e.g., click on) theobtain hedge button 940 to obtain the hedging portfolio generated by theRB hedger using the current settings. In one embodiment, the RB hedgersystem 110 automatically initiates trades with an electronic tradingsystem to obtain the assets in the hedging portfolio for the user.Alternatively, selecting the obtain hedge button 940 may display a listof assets in the hedging portfolio and open a trading interface withwhich the user can place orders to obtain the assets in the hedgingportfolio.

Computing System Architecture

FIG. 10 illustrates an example computer 1000 suitable for use as the RBhedger system 110 or a client device 140. The example computer 1000includes at least one processor 1002 coupled to a chipset 1004. Thechipset 1004 includes a memory controller hub 1020 and an input/output(I/O) controller hub 1022. A memory 1006 and a graphics adapter 1012 arecoupled to the memory controller hub 1020, and a display 1018 is coupledto the graphics adapter 1012. A storage device 1008, keyboard 1010,pointing device 1014, and network adapter 1016 are coupled to the I/Ocontroller hub 1022. Other embodiments of the computer 1000 havedifferent architectures.

In the embodiment shown in FIG. 10 , the storage device 1008 is anon-transitory computer-readable storage medium such as a hard drive,compact disk read-only memory (CD-ROM), DVD, or a solid-state memorydevice. The memory 1006 holds instructions and data used by theprocessor 1002. The pointing device 1014 is a mouse, track ball,touch-screen, or other type of pointing device, and is used incombination with the keyboard 1010 (which may be an on-screen keyboard)to input data into the computer system 1000. The graphics adapter 1012displays images and other information on the display 1018. The networkadapter 1016 couples the computer system 1000 to one or more computernetworks (e.g., network 170). The types of computers used by theentities of FIGS. 1 and 3 can vary depending upon the embodiment and theprocessing power required by the entity. Furthermore, the computers canlack some of the components described above, such as keyboards 1010,graphics adapters 1012, and displays 1018.

Additional Considerations

Some portions of above description describe the embodiments in terms ofalgorithmic processes or operations. These algorithmic descriptions andrepresentations are commonly used by those skilled in the computing artsto convey the substance of their work effectively to others skilled inthe art. These operations, while described functionally,computationally, or logically, are understood to be implemented bycomputer programs comprising instructions for execution by a processoror equivalent electrical circuits, microcode, or the like. Furthermore,it has also proven convenient at times, to refer to these arrangementsof functional operations as modules, without loss of generality.

As used herein, any reference to “one embodiment” or “an embodiment”means that a particular element, feature, structure, or characteristicdescribed in connection with the embodiment is included in at least oneembodiment. The appearances of the phrase “in one embodiment” in variousplaces in the specification are not necessarily all referring to thesame embodiment. Similarly, use of “a” or “an” preceding an element orcomponent is done merely for convenience. This description should beunderstood to mean that one or more of the element or component ispresent unless it is obvious that it is meant otherwise.

Where values are described as “approximate” or “substantially” (or theirderivatives), such values should be construed as accurate +/−10% unlessanother meaning is apparent from the context. From example,“approximately ten” should be understood to mean “in a range from nineto eleven.”

As used herein, the terms “comprises,” “comprising,” “includes,”“including,” “has,” “having” or any other variation thereof, areintended to cover a non-exclusive inclusion. For example, a process,method, article, or apparatus that comprises a list of elements is notnecessarily limited to only those elements but may include otherelements not expressly listed or inherent to such process, method,article, or apparatus. Further, unless expressly stated to the contrary,“or” refers to an inclusive or and not to an exclusive or. For example,a condition A or B is satisfied by any one of the following: A is true(or present) and B is false (or not present), A is false (or notpresent) and B is true (or present), and both A and B are true (orpresent).

Upon reading this disclosure, those of skill in the art will appreciatestill additional alternative structural and functional designs for asystem and a process for simulating the performance of a hedgingportfolio. Thus, while particular embodiments and applications have beenillustrated and described, it is to be understood that the describedsubject matter is not limited to the precise construction and componentsdisclosed. The scope of protection should be limited only by any claimsthat may ultimately issue.

What is claimed is:
 1. A method for training and using aregularization-based model to build a hedging portfolio, the methodcomprising: identifying, using a processor, a target asset; applying,using the processor, the regularization-based model to identify arecommended portfolio of assets that track the target asset, whereinapplying the regularization-based model includes: training theregularization-based model using asset price data from an observationperiod; choosing initial hyperparameters of the regularization-basedmodel using asset price data from a validation period, the validationperiod being after the observation period; identifying an initialportfolio of assets by applying the regularization-based model to assetprice data from a backtest period, the backtest period being after thevalidation period; providing for display in a user interface informationregarding the initial portfolio of assets; receiving, via the userinterface, an end-user modification to one or more hyperparameters ofthe regularization-based model to obtain updated hyperparameters;retraining the regularization-based model using the updatedhyperparameters as modified by the end-user; identifying an updatedportfolio of assets by applying the retrained regularization-based modelto the asset price data from the backtest period, wherein the updatedportfolio of assets is the recommended portfolio of assets; andproviding for display in the user interface updated informationregarding the recommended portfolio of assets; and obtaining, using theprocessor, the recommended portfolio of assets.
 2. The method of claim1, wherein the hyperparameters include a concentration hyperparameterthat impacts a total number of assets in the recommended portfolio ofassets.
 3. The method of claim 1, wherein the hyperparameters include adiversity hyperparameter that impacts a range of relative sizes of assetpositions in the recommended portfolio of assets.
 4. The method of claim1, wherein the regularization-based model uses a linear objectivefunction.
 5. The method of claim 4, wherein the linear objectivefunction is solved according to a combination of linear and quadraticconstraints.
 6. The method of claim 1, wherein the information regardingthe portfolio of assets includes one or more metrics representing aperformance of the initial portfolio and the user interface includescontrols configured to enable the end-user to provide the modificationto the one or more hyperparameters of the regularization-based model. 7.The method of claim 6, wherein the updated information includes one ormore updated metrics representing a performance of the updatedportfolio.
 8. The method of claim 6, wherein the one or more metricsinclude at least one of: a tracking error, a holding error, a dailycorrelation, a hedge transaction cost, an annual volatility, or a chartindicating relative prices of the initial portfolio and the targetasset.
 9. The method of claim 6, wherein the user interface includes oneor more asset-selection controls configured to enable the end-user toselect the target asset, and wherein identifying the target assetincludes receiving an identifier of the target asset from the clientdevice in response to end-user input using the one or moreasset-selection controls.
 10. The method of claim 1, wherein retrainingthe regularization-based model is done in real time.
 11. Anon-transitory computer-readable medium storing instructions fortraining and using a regularization-based model to build a hedgingportfolio that, when executed by a computing device, cause the computingdevice to perform operations comprising: identifying a target asset;applying the regularization-based model to identify a recommendedportfolio of assets that track the target asset, wherein applying theregularization-based model includes: training the regularization-basedmodel using asset price data from an observation period; choosinginitial hyperparameters of the regularization-based model using assetprice data from a validation period, the validation period being afterthe observation period; identifying an initial portfolio of assets byapplying the regularization-based model to asset price data from abacktest period, the backtest period being after the validation period;providing for display in a user interface information regarding theinitial portfolio of assets; receiving, via the user interface, anend-user modification to one or more hyperparameters of theregularization-based model to obtain updated hyperparameters; retrainingthe regularization-based model using the setting updated hyperparametersas modified by the end-user; identifying an updated portfolio of assetsby applying the retrained regularization-based model to the asset pricedata from the backtest period, wherein the updated portfolio of assetsis the recommended portfolio of assets; and providing for display in theuser interface updated information regarding the recommended portfolioof assets; and obtaining the recommended portfolio of assets.
 12. Thenon-transitory computer-readable medium of claim 11, wherein thehyperparameters include a concentration hyperparameter that impacts atotal number of assets in the recommended portfolio of assets.
 13. Thenon-transitory computer-readable medium of claim 11, wherein thehyperparameters include a diversity hyperparameter that impacts a rangeof relative sizes of asset positions in the recommended portfolio ofassets.
 14. The non-transitory computer-readable medium of claim 11,wherein the regularization-based model uses a linear objective function.15. The non-transitory computer-readable medium of claim 14, wherein thelinear objective function is solved according to a combination of linearand quadratic constraints.
 16. The non-transitory computer-readablemedium of claim 11, wherein the information regarding the portfolio ofassets includes one or more metrics representing a performance of theinitial portfolio and the user interface includes controls configured toenable the end-user to provide the modification to a setting of theregularization-based model.
 17. The non-transitory computer-readablemedium of claim 16, wherein the updated information includes one or moreupdated metrics representing a performance of the updated portfolio. 18.The non-transitory computer-readable medium of claim 16, wherein the oneor more metrics include at least one of: a tracking error, a holdingerror, a daily correlation, a hedge transaction cost, an annualvolatility, or a chart indicating relative prices of the initialportfolio and the target asset.
 19. The non-transitory computer-readablemedium of claim 11, wherein the user interface includes one or moreasset-selection controls configured to enable the end-user to select thetarget asset, and wherein identifying the target asset includesreceiving an identifier of the target asset from the client device inresponse to end-user input using the one or more asset-selectioncontrols.
 20. The non-transitory computer-readable medium of claim 11,wherein retraining the regularization-based model is done in real time.